Laser sustained plasma or continuous optical discharge (COD) predicted and first obtained in pioneering works initiated and inspired by Prof. Yu.P. Raizer [1-3] is still the unique method of sustaining localized dense plasma in the laboratory and some special devices like high brightness broadband radiation sources [6-10]. Although the parameters determining COD state can be stabilized, in practice COD tends to regular oscillations. As stated in the paper, the oscillations may be attributed to regular pulsing of thermal gravitational convective plume around COD. Application problems arising from instability of this type is pulsing of the emission of the promising radiation sources utilizing COD plasma. The paper represents detailed analysis of experimental data on COD convective plume radii and oscillation frequencies depending on the plasma forming gas pressure. The analysis leads to similarity relation for the oscillation frequency. The similarity relation turns out to be common for optical discharges and laminar flames under conditions of prevailing buoyancy forces. This indicates the hydrodynamic nature of the instability considered, regardless of the energy input method in the form of absorbed laser radiation or chemical reaction. To support the idea of hydrodynamic origin of the pulsations a numerical simulation of convective plume from concentrated heat source was carried out. Thermal dissipation power of the heat source was equal to that of COD. The results of numerical simulations demonstrate good agreement to those of the experiments. Dynamic temperature, density and gas velocity distributions obtained show that the direct cause of the oscillations is dynamics of toroidal vortices developed in the convection plume. Pulsing frequency equal to the frequency of vortices formation may be increased under incoming additional forced gas flow that may also suppress the convective oscillations. The results obtained may be useful for studying optical discharges and improving the parameters of high-brightness broadband laser-plasma radiation sources based on them.